The best value of \(\beta _t\)  to use depends greatly on the precise objective, however. In general, when \(\beta _{t}\)  is large, DTS behaves more like a typical bandit algorithm, allocating a higher fraction of effort to the arm a top-one TS would select. This requires running a longer experiment, which can be inherently costly due to operational considerations. However a larger \(\beta _t\)  could lead to lower regret incurred during the experiment or allow for narrower confidence intervals on the quality of the chosen arm. It would be interesting in future work ot incorporate those considerations into the objective and set \(\beta \)  accordingly. In a related model of best-arm identification, Theorem 1 of [1]} provides results for arbitrary fixed \(\beta \) : in that case, \(\frac{1}{t} \log (1- \alpha _{t,I^*}) \rightarrow -\Gamma _{\theta ,\beta }^{-1}\)  where bounds like \(\Gamma _{\theta ,0.5}^{-1} \geqslant (1/2)\Gamma _{\theta }^{-1}\)  control the worst-case impact of setting \(\beta \)  that large. It seems the same analysis could be carried out in the setting of this paper.
